Annu. Rev. Astron. Astrophys. 1997. 35: 101-136
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8.4. Simulation Approach

As noted above, Scott (1957) was a pioneer in making numerical simulations, at that time using tables of random numbers in the study of how selection effects influence the distribution function of standard candles and hence the inferred distances. More recently, computers have often been used in this manner, either to show the existence of or to illuminate some effect, for testing a correction method, or for using a Monte Carlo procedure as an integral part of the method. An advantage of such experiments, found in many of the mentioned references, is that one can construct a synthetic galaxy universe where the true distances and values of other relevant parameters are known and where imposition of the known selection effects simulates what the astronomer sees "in the sky."

Simulations have been used to show what happens in the classical Hubble diagram when the line of sight traverses a concentration of galaxies (e.g. Landy & Szalay 1992, Ekholm & Teerikorpi 1994 Section 6.5.2 in Strauss & Willick 1995). However, the most extensive applications have concerned the methods and Malmquist corrections needed when one tries to map the peculiar velocity field, especially using the POTENT algorithm (Dekel et al 1993). Indeed, Dekel et al (1993) made the classical (homogeneous) Malmquist correction to distances inferred from the direct relation, whereas Newsam et al (1995) suggest the use of the inverse relation in their iterative variant of POTENT. Strauss & Willick (1995) describe their "Method II+," which incorporates random peculiar velocities in a maximum likelihood method, applicable either to the direct or, preferably, to the inverse distance indicator. Nusser & Davis (1995) use the inverse relation in their method for deriving a smoothed estimate of the peculiar velocity field, and they support the method using simulations on a synthetic data set. Freudling et al (1995) gives several examples of how the peculiar velocity field is deformed because of unattended bias, including Gould's effect.

Clearly, theoretical understanding of how the Malmquist biases affect studies of the peculiar velocities is rapidly advancing. On the balance, it is worthwhile to be cautious of how successfully one can apply the actual inverse relation, which is favored in such theoretical studies, to the real data.

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